Around logical perfection

TL;DR

This paper introduces the concept of 'logical perfection' by linking categoricity in model theory with classical notions of mathematical robustness and smoothness, proposing a new category of perfect logical structures.

Contribution

It formalizes 'logical perfection' as a new notion derived from categoricity and classical mathematical ideas, expanding the conceptual framework of model theory.

Findings

Defines 'logical perfection' based on categoricity in power.

Connects modern model theory with classical mathematical notions.

Proposes a category of logically perfect structures.

Abstract

In this article we present and describe a notion of "logical perfection". We extract the notion of "perfection" from the contemporary logical concept of categoricity. Categoricity (in power) has become in the past half century a main driver of ideas in model theory, both mathematically (stability theory may be regarded as a way of approximating categoricity) and philosophically. In the past two decades, categoricity notions have started to overlap with more classical notions of robustness and smoothness. These have been crucial in various parts of mathematics since the nineteenth century. We postulate and present the category of logical perfection. We draw on various notions of perfection from mathematics of the 19th and 20th centuries and then trace the relation to the concept of categoricity in power as a logical notion of what a "mathematically perfect" structure is.